Question

use the riemann sum to estimate the (signed) area beneath the curve f(x)=x^2 over the interval (0,6)with 3 rectangles

Answer 1

The function is:

$f(x)=x^2$

we start by calculating the delta of x

$\Delta x=\frac{6-0}{3}=2$

Then

$\begin{gathered} x_i=x_0+i\text{ }\Delta x\text{ donde }i=1,2,3 \\ x_0=0 \\ x_1=0+1(2)=2 \\ x_2=0+2(2)=4 \\ x_3=0+3(2)=6 \end{gathered}$$\begin{gathered} f(x_i) \\ f(2)=2^2=4 \\ f(4)=4^2=16 \\ f(6)=6^2=36 \end{gathered}$

And the riemann sum is:

$\begin{gathered} A=\sum_{n\mathop{=}1}^3f(x_1)\text{ }\Delta x \\ A=(4+16+36)*2=112 \end{gathered}$